Structural Change and Economic Dynamics 11 2000 45 – 65
On detectable and non-detectable structural change
David F. Hendry
Nuffield College, New Road, Oxford OX
1 1
NF, UK
Abstract
A range of parameter changes in I1 cointegrated time series are not reflected in econometric models thereof, in that many shifts are not easily detected by conventional tests.
The breaks in question are changes that leave the unconditional expectations of the I0 components unaltered. Thus, dynamics, adjustment speeds etc. may alter without detection.
However, shifts in long-run means are generally noticeable. Using the VECM model class, the paper discusses such results, explains why they occur, and uses Monte Carlo experiments
to illustrate the contrasting ease of detection of ‘deterministic’ and ‘stochastic’ shifts. © 2000 Elsevier Science B.V. All rights reserved.
JEL classification
:
C15; C53 Keywords
:
Cointegrated time series; Parameter changes; Forecast error www.elsevier.nllocatestrueco
1. Introduction
Structural change and dynamics are inherent facets of economic life, and over recorded history have revolutionized the human condition: statistical tests are not
needed to discriminate between conditions in 1899 and 1999. Nevertheless, it transpires that a range of parameter changes in econometric models cannot be
easily detected by conventional tests, whereas other changes are manifest and easy to detect. The former class includes changes that leave unaltered the unconditional
expectations of non-integrated denoted I0 components even if dynamics, adjust- ment speeds, and intercepts are radically altered. The later class comprises shifts in
those unconditional expectations. Some related illustrations are provided in Hendry and Doornik 1997, Clements and Hendry 1998.
Corresponding author. Tel.: + 44-1865-278554; fax: + 44-1865-278557. 0954-349X00 - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 5 4 - 3 4 9 X 0 0 0 0 0 2 0 - 5
This paper extends those studies, using a co-integrated vector autoregression CVAR as the general model. The analytical framework is the taxonomy of
possible sources of forecast error developed in Clements and Hendry 1994, which highlights the impact of various forms of structural break in closed dynamic
systems, and allows some surprising predictions about which parameters of a VAR can be altered without much impact on the properties of the resulting data. In
matching Monte Carlo experiments, a bivariate system is subject to two structural breaks in its parameters, where the second re-instates the original parameter values,
to show the contrast between the ‘detectable’ and relatively ‘non-detectable’ changes.
1
For example, even when the general data are integrated of order unity I1, the intercepts and the dynamic coefficients of a CVAR can be changed
considerably i.e. by more than 50, yet parameter-constancy tests do not detect such changes. Moreover, this is not due to model mis-specification: the in-sample
model considered here coincides with the data generation process DGP. Nor is it a matter of generically low-powered tests: the same tests have high power to detect
other shifts in either intercepts or dynamic coefficients. The paper explains why this outcome occurs: for more general studies of the size and power properties of
parameter-constancy tests in both I0 and I1 systems, see Anderson et al. 1993, Hylleberg et al. 1993.
To summarize why non-detectability occurs, let y
t
denote a vector I0 time series
with actual unconditional expectations and variances denoted by E[y
t
] and V[y
t
] respectively. Let the corresponding expectations, based on assuming the model is
the in-sample DGP, be denoted by E[y
t
] and V[y
t
] an alternative notation might be E
M
[·]. Then so long as V[y
t
] is not markedly different from V[y
t
], detectability
depends strongly on the difference E[y
t
] − E[y
t
]. Consequently, parameter changes
in the class that leave E[y
t
] E[y
t
], suffer from a detectability problem unless they generate large variance increases. Since I1 CVARs can be reparameterized by
differencing and cointegration transformations as I0 vector equilibrium-correction models VEqCMs, where all variables are expressed as deviations around their
pre-break means, the same logic applies: only shifts in those means induce departures that are readily detectable. Indeed, it is easy to create major parameter
shifts that leave the first two moments virtually unchanged, and such breaks will be almost impossible to detect using tests based on those moments. Although other
aspects of the model might reveal such shifts, including residual auto-correlation when dynamics alter, the correct interpretation would not be obvious, and indeed
other tests could well be distorted by the mis-specification deriving from the break. Different tests might be developed to prise out the correct underlying state of
nature, but here we focus on the properties of recursively-computed F-tests of parameter constancy, and the associated recursive parameter estimates.
The structure of the paper is as follows. Section 2 describes the n-dimensional I1 DGP that forms the class within which the analysis is undertaken. Section 3
1
Of course, for a sufficiently large sample, both pre and post any change, and for a big enough break, all shifts are detectable on the tests used here. The practical issue, however, is their ease of detection for
realistic sample sizes and breaks of a plausible magnitude.
formalizes the structural breaks to be studied, and Section 3.1 discusses which changes are and are not easily detected. This is followed by a Monte Carlo study
of a bivariate I1 CVAR in Section 4, for three classes of structural break, and in Section 4.5 for shifts in the cointegration space. Section 5 offers potential explana-
tions for the mixture of results obtained and Section 6 concludes.
2. The data generation process Consider a first-order vector autoregression in n variables denoted x
